Suppose the firm uses two inputs to produce its output: capital (K) and labor (L). The firm’s
production function is . Suppose that in the short run capital (K) is fixed at 27. If the
wage is and the rental rate on capital is ,

Does the production function display decreasing, constant, or increasing returns to
scale? Explain

What is the short run total cost equation (function with ?

What is the level of output (q) that minimizes the total cost?

Ask by Gibbs Mcguire. in Palestine

Jan 07,2025

Question

Suppose the firm uses two inputs to produce its output: capital (K) and labor (L). The firm’s
production function is . Suppose that in the short run capital (K) is fixed at 27. If the
wage is and the rental rate on capital is ,

Does the production function display decreasing, constant, or increasing returns to
scale? Explain

What is the short run total cost equation (function with ?

What is the level of output (q) that minimizes the total cost?

Ask by Gibbs Mcguire. in Palestine

Jan 07,2025

UpStudy AI · Accepted Answer

Let's address each question step by step.

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### **1. Returns to Scale of the Production Function**

**Production Function:** $$ q = L^{1/2} K^{1/3} $$

**Understanding Returns to Scale:**  
Returns to scale examine how output changes when all inputs are scaled by a common factor. Specifically:

- **Increasing Returns to Scale:** Output increases by a greater proportion than the increase in inputs.
- **Constant Returns to Scale:** Output increases by the same proportion as the increase in inputs.
- **Decreasing Returns to Scale:** Output increases by a smaller proportion than the increase in inputs.

**Analysis:**
- **Scale Inputs by a Factor $$ t $$:**  
  $$ q' = (tL)^{1/2} (tK)^{1/3} = t^{1/2} L^{1/2} \cdot t^{1/3} K^{1/3} = t^{5/6} \cdot q $$

- **Compare Output Scaling:**  
  Since $$ t^{5/6} < t $$ for $$ t > 1 $$, the output increases by a smaller proportion than the increase in inputs.

**Conclusion:**  
The production function exhibits **decreasing returns to scale** because scaling both inputs by a factor $$ t $$ results in output scaling by $$ t^{5/6} $$, which is less than $$ t $$.

---

### **2. Short-Run Total Cost Equation**

**Given:**
- **Fixed Capital (K):** 27
- **Wage Rate (w):** \$4.5 per unit of labor (L)
- **Rental Rate of Capital (r):** \$4 per unit of capital

**Total Cost Components:**
- **Fixed Cost (FC):**  
  $$ FC = r \times K = 4 \times 27 = \$108 $$

- **Variable Cost (VC):**  
  $$ VC = w \times L = 4.5 \times L $$

**Expressing Labor (L) in Terms of Output (q):**
- **Production Function in Short Run:**  
  $$ q = L^{1/2} \times 27^{1/3} $$

- **Simplify:**  
  $$ 27^{1/3} = 3 $$  
  $$ q = 3 L^{1/2} $$  
  $$ L^{1/2} = \frac{q}{3} $$  
  $$ L = \left( \frac{q}{3} \right)^2 = \frac{q^2}{9} $$

- **Variable Cost in Terms of q:**  
  $$ VC = 4.5 \times \frac{q^2}{9} = 0.5 q^2 $$

**Total Cost (TC) Equation:**  
$$ TC(q) = FC + VC = 108 + 0.5 q^2 $$

---

### **3. Output Level that Minimizes Total Cost**

**Total Cost Function:**  
$$ TC(q) = 108 + 0.5 q^2 $$

**Objective:**  
Find the level of output $$ q $$ that minimizes $$ TC(q) $$.

**Analysis:**
- The function $$ TC(q) = 108 + 0.5 q^2 $$ is a quadratic function in terms of $$ q $$ with a positive coefficient for $$ q^2 $$, which means it has a **minimum** at $$ q = 0 $$.

- **However, producing zero output is trivial and typically not meaningful in practical scenarios.** Instead, economic problems often interpret such questions as minimizing **average total cost (ATC)** rather than total cost itself.

**Minimizing Average Total Cost (ATC):**  
- **Average Total Cost:**  
  $$ ATC(q) = \frac{TC(q)}{q} = \frac{108}{q} + 0.5 q $$

- **To Find Minimum ATC:**  
  Take the derivative of $$ ATC(q) $$ with respect to $$ q $$ and set it to zero:  
  $$
  \frac{d(ATC)}{dq} = -\frac{108}{q^2} + 0.5 = 0 \
  \Rightarrow 0.5 = \frac{108}{q^2} \
  \Rightarrow q^2 = \frac{108}{0.5} = 216 \
  \Rightarrow q = \sqrt{216} = 6\sqrt{6} \approx 14.7
  $$

**Conclusion:**  
While the total cost is technically minimized at $$ q = 0 $$, the more economically meaningful interpretation is that the **average total cost is minimized when the output level $$ q $$ is approximately 14.7** units.

---

### **Summary of Answers**

1. **Decreasing Returns to Scale:** Scaling both inputs by a factor $$ t $$ increases output by $$ t^{5/6} $$, which is less than proportional, indicating decreasing returns to scale.
2. **Short-Run Total Cost Equation:**  
   $$ TC(q) = 108 + 0.5 q^2 $$
3. **Output Level Minimizing Cost:**  
   $$ q \approx 14.7 $$ units minimizes the average total cost.

---
1. The production function has **decreasing returns to scale** because increasing both inputs by a factor $$ t $$ results in output increasing by less than $$ t $$. 2. The short-run total cost equation is: $$ TC(q) = 108 + 0.5q^2 $$ 3. The output level that minimizes the average total cost is approximately **14.7 units**.

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